Singular stochastic control in the presence of a state-dependent yield structure

Alvarez, Luis H. R.

doi:10.1016/s0304-4149(99)00102-7

Cited by 47 publications

(50 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Mathematical analysis of such control problems has evolved considerably from the initial heuristics to the more sophisticated and standard stochastic control approach, and from the very special case study to general payoff functions. (See Harrison and Taksar (1983); Karatzas (1985); Karatzas and Shreve (1985); El Karoui and Karatzas ( , 1989; Ma (1992); Zervos (1994, 1998); Boetius and Kohlmann (1998); Alvarez (2000Alvarez ( , 2001; Bank (2005); Boetius (2005)). Most recently, Merhi and Zervos (2007) analyzed this problem in great generality and provided explicit solutions for the special case where the payoff is of Cobb-Douglas type.…”

Section: Introductionmentioning

confidence: 99%

A Class of Singular Control Problems and the Smooth Fit Principle

Guo¹,

Tomecek

2009

SIAM J. Control Optim.

View full text Add to dashboard Cite

This paper analyzes a class of singular control problems for which value functions are not necessarily smooth. Necessary and sufficient conditions for the well-known smooth fit principle, along with the regularity of the value functions, are given. Explicit solutions for the optimal policy and for the value functions are provided. In particular, when payoff functions satisfy the usual Inada conditions, the boundaries between action and no-action regions are smooth and strictly monotonic as postulated and exploited in the existing literature (Dixit and Pindyck (1994)

show abstract

Section: Introductionmentioning

confidence: 99%

A Class of Singular Control Problems and the Smooth Fit Principle

Guo¹,

Tomecek

2009

SIAM J. Control Optim.

View full text Add to dashboard Cite

show abstract

“…This heuristic argument -also discussed at pp. 329-330 of [2] in the context of one-dimensional monotone follower problems -can be rigorously justified, and technical details on the convergence can be found in the recent [5]. We also refer to [17] and [29] as other papers on singular stochastic control problems employing such a definition for the integral with respect to the control process.…”

Section: Setting and Problem Formulationmentioning

confidence: 99%

“…Step 2. To prove the monotonicity of V with respect to σ we adapt to our setting ideas from Theorem 4 in [2]. Let V be the value function when the volatility coefficient in (2.2) iŝ σ.…”

Section: Sensitivity Analysis In the Case Of A Drifted Brownian Motiomentioning

confidence: 99%

An Optimal Extraction Problem with Price Impact

Ferrari

Koch

2019

Appl Math Optim

View full text Add to dashboard Cite

A price-maker company extracts an exhaustible commodity from a reservoir, and sells it instantaneously in the spot market. In absence of any actions of the company, the commodity's spot price evolves either as a drifted Brownian motion or as an Ornstein-Uhlenbeck process. While extracting, the company affects the market price of the commodity, and its actions have an impact on the dynamics of the commodity's spot price. The company aims at maximizing the total expected profits from selling the commodity, net of the total expected proportional costs of extraction. We model this problem as a two-dimensional degenerate singular stochastic control problem with finite fuel. To determine its solution, we construct an explicit solution to the associated Hamilton-Jacobi-Bellman equation, and then verify its actual optimality through a verification theorem. On the one hand, when the (uncontrolled) price is a drifted Brownian motion, it is optimal to extract whenever the current price level is larger or equal than an endogenously determined constant threshold. On the other hand, when the (uncontrolled) price evolves as an Ornstein-Uhlenbeck process, we show that the optimal extraction rule is triggered by a curve depending on the current level of the reservoir. Such a curve is a strictly decreasing C ∞ -function for which we are able to provide an explicit expression. Finally, our study is complemented by a theoretical and numerical analysis of the dependency of the optimal extraction strategy and value function on the model's parameters.

show abstract

“…(2) As discussed in the introduction, problems of optimal harvesting or of optimal dividends' distribution with (compulsory) capital injections might be modeled as (2.21). More in detail, problem (2.21) might be interpreted as an optimal harvesting problem (see, e.g., [3] and [4]), in which the harvester has the obligation of preserving a population that can extinguish in finite time. In this case, X D is the (logarithm of the) size of the population, D t is the cumulative amount of members of the population that have been harvested up to time t ≥ 0, and L D t the cumulative amount of members of the species that have been reintroduced up to time t ≥ 0 in order to avoid extinction.…”

Section: 2mentioning

confidence: 99%

“…Two problems from applications leading to (1.1) are those of optimal dividends' distribution with capital injections (see, e.g., [27], [30], and [46]), and of optimal harvesting (see, e.g., [3], [4], and references therein).…”

Section: Introductionmentioning

confidence: 99%

On a class of singular stochastic control problems for reflected diffusions

Ferrari

2019

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Reflected diffusions naturally arise in many problems from applications ranging from economics and mathematical biology to queueing theory. In this paper we consider a class of infinite time-horizon singular stochastic control problems for a general onedimensional diffusion that is reflected at zero. We assume that exerting control leads to a state-dependent instantaneous reward, whereas reflecting the diffusion at zero gives rise to a proportional cost with constant marginal value. The aim is to maximize the total expected reward, minus the total expected cost of reflection. We show that depending on the properties of the state-dependent instantaneous reward we can have qualitatively different kinds of optimal strategies. The techniques employed are those of stochastic control and of the theory of linear diffusions.

show abstract

Singular stochastic control in the presence of a state-dependent yield structure

Cited by 47 publications

References 11 publications

A Class of Singular Control Problems and the Smooth Fit Principle

A Class of Singular Control Problems and the Smooth Fit Principle

An Optimal Extraction Problem with Price Impact

On a class of singular stochastic control problems for reflected diffusions

Contact Info

Product

Resources

About